Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond percolation on $\mathcal{C}_p$ parametrized by $p$ (or parametrized by a $q\ne p$)? Phase caracterizations? Critical values $q_c=f(p)$? Number of infinite clusters if any?
